Acknowledgement
The authors are grateful to the reviewers for their valuable remarks and advices that help us to improve the quality of the paper in present form.
References
- M. Abramowitz and I.A. Stegun (Eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., New York, 1965.
- A. Attiya, Some Applications of Mittag-Leffler Function in the Unit Disk, Filomat, 30(7) (2016), 2075-2081. https://doi.org/10.2298/FIL1607075A
- D. Bansal and J.K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61 (3) (2016), 338-350. https://doi.org/10.1080/17476933.2015.1079628
- D.A. Brannan, J. Clunie and W.E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22(3) (1970), 476-485. https://doi.org/10.4153/CJM-1970-055-8
- D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math., 31(2) (1986), 70-77.
- T. Bulboaca, Differential Subordinations and Superordinations. Recent Results, House of Scientific Book Publ., Cluj-Napoca, (2005).
- M. Caglar, H. Orhan and N. Yagmur, Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27 (2013), 1165-1171. https://doi.org/10.2298/FIL1307165C
- E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal., 2(1) (2013), 49-60. https://doi.org/10.7153/jca-02-05
- P.L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
- A. Ebadian, N.E. Cho, E.A. Adegani, S. Bulut and T. Bulboaca, Radii problems for some classes of analytic functions associated with Legendre polynomials of odd degree, JIA (2020). 178,https://doi.org/10.1186/s13660-020-02443-4.
- S.M. El-Deeb, Maclaurin coefficient estimates for new subclasses of bi-univalent functions connected with a q-analogue of Bessel function, Abstract Appl. Anal., Article ID 8368951, (2020), 1-7, https://doi.org/10.1155/2020/8368951.
- S.M. El-Deeb, T. Bulboaca and B.M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative, Mathematics, 8 (2020), 1-14,https://doi.org/10.3390/math8030418.
- S.M. El-Deeb and B.M. El-Matary, New subclasses of bi-univalent functions connected with a q-analogue of convolution based upon the Legendre polynomials, Stud. Univ. Babes-Bolyai Math.,(accepted for publications-2021).
- B.A. Frasin, An application of an operator associated with generalized Mittag-Leffler function, Konuralp J. Math.,7(1) (2019), 199-202.
- B.A. Frasin, T. Al-Hawary and F. Yousef, Some properties of a linear operator involving generalized Mittag-Leffler function,Stud. Univ. Babes-Bolyai Math., 65(1) (2020), 67-75. https://doi.org/10.24193/subbmath.2020.1.06
- M. Garg, P. Manohar and S.L. Kalla, A Mittag-Leffler-typefunction of two variables, Integral Transforms Spec. Funct., 24(11) (2013), 934-944. https://doi.org/10.1080/10652469.2013.789872
- G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, 2004.
- F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(2) (1909), 253-281. https://doi.org/10.1017/S0080456800002751
- F.H. Jackson, q- difference equations, Amer. J. Math., 32(4) (1910), 305-314. https://doi.org/10.2307/2370183
- V. Kac and P. Cheung, Quantum calculus, Springer-Verlag, New-York, 2002.
- V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics, Vol. 301, Longman Scientific and Technical, Harlow (Essex), 1993.
- N. Lebedev, Special functions and their applications, Dover, New York, (1972).
- M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63-68. https://doi.org/10.1090/S0002-9939-1967-0206255-1
- S.S. Miller and P.T. Mocanu, Differential Subordinations. Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, 2000.
- G.M. Mittag-Leffler, Sur la nouvelle fonction E(x), C. R. Acad. Sci. Paris, 137 (1903), 554-558.
- G. Murugusundaramoorthy and S.M. El-Deeb, Second Hankel determinant for a class of analytic functions of the Mittag-Leffler-type Borel distribution related with Legendre polynomials, Turkish World Math. Soc. J. Appl. Engineering Math., (accepted for publications-2021).
- Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952.
- H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10) (2010), 1188-1192. https://doi.org/10.1016/j.aml.2010.05.009
- A.K. Wanas and J.A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline J. Math. Sci., 4(1) (2020), 71-82.
- A. Wiman, Uber die Nullstellun der Funcktionen E(x), Acta Math., 29 (1905), 217-134. https://doi.org/10.1007/BF02403204